Here are two more translations of the same riddle, the first by Elizabeth Hickman du Bois:

My name is Greek, for me great beauties vied,A fair youth's fraud, with sisters I abide,Through me was vanquished mighty Ilion's pride.

and the second by Raymond Theodore Ohl:

The name for sheep in Greek, the cause of great strife among the goddesses, the guile of the girded youth, the care of many sisters, the destruction of Troy what time I brought to an end bloody wars.

I haven't seen Raymond Theodore Ohl, The Enigmas of Symphosius (Philadephia, 1928); I borrowed Ohl's translation from Bill Thayer's web pages on Symphosius.

There are riddles within the riddle.

First, Nomen ovis Graece ("the name for sheep in Greek"). In Greek, there are two different words both spelled μῆλον (mēlon). One of the homonyms means sheep (in Latin ovis), the other apple (in Latin mālum). Wilbur's "My Latin name sounds evil" refers to the fact that the Latin adjective mălus, -a, -um means evil, differing from the noun mālum (apple) only by the quantity of the -a-.

Plautus, Amphitruo 720-724 (tr. Paul Nixon), puns on the two similar Latin words for evil (mălum) and apple (mālum):

ALC. Indeed I have not, and I pray heaven I may safely bear a son. But you, sir, shall have an ample supply of aches and pains, if your master here does his duty! You shall be well rewarded for that omen, Sir Omener.

SOS. Really now, ma'am, it's a lady in your condition ought to have aches and pains, yes, and an apple supply, too, so as to have something to chew on in case she gets to feeling seedy.

To the modern reader, Wilbur's "My Latin name sounds evil" also brings to mind the forbidden fruit of Genesis, chapters 2-3, which St. Jerome translated into Latin as mālum. The eating of the apple (mālum), in violation of God's command, brought evil (mălum) into the world. See Horace Jeffery Hodges, "Forbidden Fruit as Impedimental Peach: A Scholarly 'Pesher' on Paradise Lost 9.850-852," Milton and Early Modern English Studies 18.2 (2008), for other puns surrounding the forbidden fruit, which some think was not an apple but a peach (French pêche, cf. French péché = sin). I am indebted to Dr. Hodges for a copy of his article.

But Symphosius was apparently a pagan, and there are no Biblical echoes in his riddle. There is a reference to the golden apple which ultimately caused the Trojan War. Lucian, Dialogues of the Sea-Gods 5 (tr. H.W. and F.G. Fowler), tells the beginning of the tale:

PANOPE. Galene, did you see what Eris [Strife] did yesterday at the Thessalian banquet, because she had not had an invitation?

GALENE. No, I was not with you; Posidon had told me to keep the sea quiet for the occasion. What did Eris do, then, if she was not there?

PANOPE. Thetis and Peleus had just gone off to the bridal chamber, conducted by Amphitrite and Posidon, when Eris came in unnoticed  which was easy enough; some were drinking, some dancing, or attending to Apollo's lyre or the Muses' songs  Well, she threw down a lovely apple, solid gold, my dear; and there was written on it, FOR THE FAIR. It rolled along as if it knew what it was about, till it came in front of Hera, Aphrodite, and Athene. Hermes picked it up and read out the inscription; of course we Nereids kept quiet; what should we do in such company? But they all made for it, each insisting that it was hers; and if Zeus had not parted them, there would have been a battle. He would not decide the matter himself, though they asked him to. 'Go, all of you, to Ida,' he said, 'to the son of Priam [Paris]; he is a man of taste, quite capable of picking out the beauty; he will be no bad judge.'

Isocrates 10.41-42 (tr. George Norlin) picks up the thread of the story:

For not much later when strife arose among the goddesses for the prize of beauty, and Alexander [Paris], son of Priam, was appointed judge and when Hera offered him sovereignty over all Asia, Athena victory in war, and Aphrodite Helen as his wife, finding himself unable to make a distinction regarding the charms of their persons, but overwhelmed by the sight of the goddesses, Alexander, compelled to make a choice of their proffered gifts, chose living with Helen before all else.

The Judgment of Paris, of course, started the Trojan War, when Helen's husband Menelaus wanted her back.

What about the trick of the fair youth (fraus iuvenis pulchri) in Symphosius' riddle? Probably this refers to the story of Atalanta. See Apollodorus 3.9.2 (tr. J.G. Frazer):

Grown to womanhood, Atalanta kept herself a virgin, and hunting in the wilderness she remained always under arms. The centaurs Rhoecus and Hylaeus tried to force her, but were shot down and killed by her. She went moreover with the chiefs to hunt the Calydonian boar, and at the games held in honor of Pelias she wrestled with Peleus and won. Afterwards she discovered her parents, but when her father would have persuaded her to wed, she went away to a place that might serve as a racecourse, and, having planted a stake three cubits high in the middle of it, she caused her wooers to race before her from there, and ran herself in arms; and if the wooer was caught up, his due was death on the spot, and if he was not caught up, his due was marriage. When many had already perished, Melanion came to run for love of her, bringing golden apples from Aphrodite, and being pursued he threw them down, and she, picking up the dropped fruit, was beaten in the race. So Melanion married her.

In other words, Atalanta ran a hoplitodromos, a foot race in armor, like the competitors in the Olympic games. If she caught her competitor in the race, she slew him. If she lost the race, she gave her hand in marriage to the winner. Melanion (aka Milanion or Hippomenes) distracted Atalanta by throwing golden apples and so won the race.

Finally, what does Symphosius mean by multarum cura sororum (care of many sisters)? Probably the sisters are the Hesperides,

the famous guardians of the golden apples which Ge had given to Hera at her marriage with Zeus. Their names are Aegle, Erytheia, Hesperia, and Arethusa, but their descent is not the same in the different traditions; sometimes they are called the daughters of Night or Erebus (Hes. Theog. 215; Hygin. Fab. init.), sometimes of Phorcys and Ceto (Schol. ad Apollon. Rhod. iv.1399), sometimes of Atlas and Hesperis, whence their names Atlantides or Hesperides (Diod. iv.27), and sometimes of Hesperus, or of Zeus and Themis. (Serv. ad Aen. iv.484; Schol. ad Eurip. Hipp. 742.) Instead of the four Hesperides mentioned above, some traditions know only of three, viz. Hespere, Erytheis, and Aegle, or Aegle, Arethusa, and Hesperusa or Hesperia (Apollon. Rhod. iv.1427; Serv. l.c.; Stat. Theb. ii.281); whereas others mention seven. (Diod. l.c.; Hygin. Fab. init.) The poets describe them as possessed of the power of sweet song. (Hes. Theog. 518; Orph. Fragm. 17; Eurip. Herc. Fur. 394; Apollon. Rhod. iv.1399.) In the earliest legends, these nymphs are described as living on the river Oceanus, in the extreme west (Hes. Theog. 334, &c., 518; Eurip. Hipp. 742); but the later attempts to fix their abodes, and the geographical position of their gardens, have led poets and geographers to different parts of Libya, as in the neighbourhood of Cyrene, Mount Atlas, or the islands on the western coast of Libya (Plin. H.N. vi.31, 36; Virg. Aen. iv.480; Pomp. Mela, iii.10), or even to the northern extremity of the earth, beyond the wind Boreas, among the Hyperboreans. In their watch over the golden apples they were assisted or superintended by the dragon Ladon.

William Smith, A Dictionary of Greek and Roman Biography and Mythology, s.v. Hesperides.